UNIVERSIDADE ESTADUAL DE CAMPINAS
SISTEMA DE BIBLIOTECAS DA UNICAMP
REPOSITÓRIO DA PRODUÇÃO CIENTIFICA E INTELECTUAL DA UNICAMP
Versão do arquivo anexado / Version of attached file:
Versão do Editor / Published Version
Mais informações no site da editora / Further information on publisher's website:
https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-019-7545-2
DOI: 10.1140/epjc/s10052-019-7545-2
Direitos autorais / Publisher's copyright statement:
©2019 by Springer. All rights reserved.
DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo
CEP 13083-970 – Campinas SP Fone: (19) 3521-6493
https://doi.org/10.1140/epjc/s10052-019-7545-2 Regular Article - Theoretical Physics
Forward elastic scattering: dynamical gluon mass and semihard
interactions
M. Broilo1,2,a, D. A. Fagundes3,b, E. G. S. Luna2,4,c, M. J. Menon5,d
1Instituto de Física e Matemática, Universidade Federal de Pelotas, Pelotas, RS 96010-900, Brazil
2Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, Porto Alegre, RS 91501-970, Brazil
3Departamento de Ciências Exatas e Educação, Universidade Federal de Santa Catarina, Campus Blumenau, Blumenau, SC 89065-300, Brazil 4Instituto de Física, Facultad de Ingeniería, Universidad de la República, J.H. y Reissig 565, 11000 Montevideo, Uruguay
5Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, Campinas, SP 13083-859, Brazil
Received: 16 October 2019 / Accepted: 6 December 2019 / Published online: 27 December 2019 © The Author(s) 2019
Abstract The role of low-x parton dynamics in dictating the high-energy behavior of forward scattering observables at LHC energies is investigated using a QCD-based model with even-under-crossing amplitude dominance at high-energies. We explore the effects of different sets of pre- and post-LHC
fine-tuned parton distributions on the forward quantitiesσt ot
andρ, from pp and ¯p p scattering in the interval 10 GeV–13
TeV. We also investigate the role of the leading soft contri-bution, the low-energy cuttoff, and the energy dependence of the semihard form factor on these observables. We show that in all cases investigated the highly restrictive data on
ρ parameter at√s = 13 TeV indicate that a crossing-odd
component may play a crucial role in forward elastic scat-tering at the highest energies, namely the Odderon contribu-tion.
1 Introduction
The elastic hadronic scattering at high energies represents a rather simple kinematic process. However, its complete dynamical description is still a fundamental problem in QCD, since the confinement phenomena precludes a pure perturba-tive approach. Over the past few years, the LHC has released precise measurements of elastic proton–proton scattering which has become an important guide for selecting models and theoretical approaches, looking for a better understand-ing of the theory of strong interactions.
ae-mail:[email protected] be-mail:[email protected] ce-mail:[email protected] de-mail:[email protected]
Among other physical observables, two forward quanti-ties play a fundamental role in the investigation of the elastic
scattering at high energies, the total cross section and theρ
parameter, which can be expressed in terms of the scattering
amplitudeA(s, t) by
σt ot(s) = 4πIm A(s, t = 0), (1)
ρ(s) = ReA(s, t = 0)
ImA(s, t = 0), (2)
where s and t are the Mandelstam variables and t = 0
indi-cates the forward direction.
Recently, the TOTEM Collaboration has provided new
experimental measurements onσt otandρ from LHC13, the
highest energy reached in accelerators. In a first paper [1], by
using as inputρ = 0.10, the measurement of the total cross
section yielded σt ot = 110.6 ± 3.4 mb.
In a subsequent work [2], an independent measurement of the total cross section was reported,
σt ot = 110.3 ± 3.5 mb,
together with the first measurements of theρ parameter:
ρ = 0.10 ± 0.01 and ρ = 0.09 ± 0.01.
Although the values of σt ot are in consensus with the
increase of previous measurements by TOTEM, theρ values
indicate a rather unexpected decrease, as compared with mea-surements at lower energies and predictions from the wide majority of phenomenological models. This new information has originated a series of recent papers and discussions on possible phenomenological explanations for the rather small ρ-value. The main concern in these theoretical discussions is the full understanding of the Odderon concept (a crossing
odd color-singlet with at least three gluons) [3–5] and of the Pomeron one (a crossing even color-singlet with at least two gluons) [6,7].
The variety of recent phenomenological analyses treats different aspects involved, pointing to distinct scenarios, and might be grouped in some classes according to their main characteristics:
• Maximal Odderon (e.g., Martynov, Nicolescu [8–11]) and Odderon effects in elastic hadron scattering (e.g., Csörg˝o, Pasechnik, Ster [12,13], Gonçalves, Silva [14]);
• discussions on Odderon effects in other reactions (e.g.,
Harland-Lang, Khoze, Martin, Ryskin [15], Gonçalves [16]);
• Pomeron dominance with small Odderon contribution
(e.g., Khoze, Martin, Ryskin [17], Gotsman, Levin,
Potashnikova [18,19], Lebiedowicz, Nachtmann,
Szczurek [20], Bence, Jenkovszky, Szanyi [21]);
• leading Pomeron without Odderon contribution in elastic
scattering (e.g., Shabelski, Shuvaev [22], Broilo, Luna, Menon [23–25], Durand and Ha [26], Donnachie and Landshoff [27]) and in other reactions (e.g., Lebiedowicz, Nachtmann, Szczurek [28]);
• reanalyzes of the differential cross section data from
TOTEM [2], indicating results forσt ot andρ at 13 TeV
different from the afore-quoted values (e.g., Pacetti, Sri-vastava, Pancheri [29], Kohara, Ferreira, Rangel [30], Cudell, Selyugin [31]).
In this rather intricate scenario, we present here a
phe-nomenological study on the forward pp and ¯p p elastic
scat-tering data in the region 10 GeV–13 TeV. In our model the
behavior of the forward quantitiesσt ot(s) and ρ(s), given
by Eqs. (1) and (2), are expected to be asymptotically dom-inated by the so-called semihard interactions. This type of process originates from hard scattering of partons which carry a very small fraction of the momenta of their parent hadrons, leading to the appearance of minijets [32,33]. The latter can be viewed simply as jets with transverse energy much smaller than the total center-of-mass energy avail-able in the hadronic collision. The energy dependence of the cross sections is driven mainly by semihard elementary processes that include at least one gluon in the initial state, since at low x they are responsible for the dominant contri-bution.
In the QCD-based formalism these partonic processes are written by means of the standard QCD cross sec-tions convoluted with partonic distribution funcsec-tions. How-ever, these processes are potentially divergent at low trans-ferred momenta, and for this reason they must be regu-larized by means of some cutoff procedure. In a nonper-turbative QCD context, one natural regulator was intro-duced by Cornwall some time ago [34], and since then has
become an important feature in eikonalized models [35– 39]. This regularization process is based on the increas-ing evidence that the gluon may develop a momentum-dependent mass, which introduces a natural scale able to separate the perturbative from the nonperturbative QCD region.
Thus, taking into account the possibility that the infrared properties of QCD can, in principle, generate an effective gluon mass, we explore the nonperturbative aspects of QCD in order to describe the total cross section and the ratio of the real-to-imaginary parts of the forward elastic scattering
amplitude in pp and ¯p p collisions. Most importantly, two
components are considered in our eikonal representation, one associated with the semihard interactions and calculated from QCD and a second one associated with soft contributions and based on the Regge-Gribov phenomenology. Except for an odd under crossing Reggeon contribution, necessary to
dis-tinguish between pp and ¯p p scattering at low energies, all
the dominant components at high energies (soft and semi-hard) are associated with even under crossing contributions, namely we have Pomeron dominance and absence of Odd-eron.
Very recently a detailed study ofσt otandρ using this for-malism has been presented in [40]. This study, using the CT14 parton distribution functions (PDFs) from a global analysis by the CTEQ-TEA group [41], has shown that, despite an overall satisfactory description of the forward data is obtained, there is evidence that the introduction of an odd-under-crossing term in the scattering amplitude may
improve the agreement with the ρ data at√s = 13 TeV.
Here we have extended the previous study in several sig-nificant ways. First, we have investigated the effects of dif-ferent updated sets of PDFs beyond the CT14 [41], namely CTEQ6L [42] and MMHT [43]. In this way we can compare the results obtained by using pre- and post-LHC fine-tuned PDFs. In addition, we discuss the effects of the low-energy cuttoff, the energy dependence of the semihard form
fac-tor (see Eqs. (14–16) below) on the behavior ofσt ot andρ
and the role of the soft interaction at high energies. We also provide explicit formulas and all the details related to the formalism.
The work is organized as follows. In Sect.2a short review
on the concept of the dynamical gluon mass is presented. In
Sect. 3 we introduce all the inputs and details concerning
our QCD-based model and in Sect. 4 we specify the data
set and the fit procedures. In Sect.5the fit results are
pre-sented, followed by a discussion on the corresponding phys-ical interpretations and implications. Our conclusions and
final remarks are the contents of Sect.6. Details on the
ana-lytical parametrization for the partonic cross section are pre-sented in “Appendix A” and comments on data-set assump-tions in “Appendix B”.
2 The dynamical gluon mass
As pointed out in the previous section, scattering amplitudes of partons in QCD contain infrared divergences. One pro-cedure to regulate this behavior is by means of a dynami-cal mass generation mechanism which is based on the fact that the nonperturbative dynamics of QCD may generate an
effective momentum-dependent mass Mg(Q2) for the
glu-ons, while preserving the local SU(3)cinvariance [44–46].
The dynamical mass Mg(Q2) introduces a natural
nonper-turbative scale and is linked to a finite infrared QCD effective
charge¯αs(Q2). The existence of a dynamical gluon mass is
strongly supported by QCD lattice results. More specifically, lattice simulations reveal that the gluon propagator is finite in the infrared region [47–54] and this result corresponds, from the Schwinger–Dyson formalism, to a massive gluon [34,55–59]. It is worth mentioning that infrared-finite QCD couplings are quite usual in the literature (for a recent review, see [60]). In addition to the evidence already mentioned in the lattice QCD, a finite infrared behavior ofαs(Q2) has been suggested, for example, in studies using QCD functional methods [61–63], and in studies of the Gribov–Zwanziger scenario [64–66].
Since the gluon mass generation is a purely dynamical effect, a formal continuum approach for tackling this non-perturbative phenomenon is provided by the aforementioned Schwinger–Dyson equations that govern the dynamics of all QCD Green’s functions [34,55–59,67,68]. These equations constitute an infinite set of coupled nonlinear integral equa-tions and, after a proper truncation procedure, it is possible to obtain as a solution an infrared finite gluon propagator, while preserving the gauge invariance (or the BRST symmetry) in
question. In this work we adopt the functional forms of Mg
and¯αs obtained by Cornwall [34] via the pinch technique in
order to derive a gauge invariant Schwinger–Dyson equation for the gluon propagator and the triple gluon vertex:
Mg2(Q2) = m2g ⎡ ⎣ln Q2+ 4Mg2(Q2 /2 ln 4m2 g/2 ⎤ ⎦ −12/11 , (3) ¯αs(Q2) = 4π β0ln Q2+ 4M2 g(Q2) /2, (4)
where is the QCD scale parameter, β0= 11 − 2nf/3 (nf
is the number of flavors) and mgis the gluon mass scale to be
phenomenologically adjusted in order to yield well founded results in strongly interacting processes. Note that the dynam-ical mass Mg2(Q2) vanishes in the limit Q2 2. It is thus
evident that in this same limit the effective charge ¯αs(Q2)
matches with the one-loop perturbative coupling:
¯αs(Q2 2) ∼
4π β0ln(Q2/2)
= αp QC D
s (Q2). (5)
In the limit Q2 → 0, in turn, the effective charge ¯αs(Q2)
have an infrared fixed point, i.e. the dynamical mass tames the
Landau pole. More precisely, if the relation mg/ > 1/2 is
satisfied then ¯αs(Q2) is holomorphic (analytic) on the range 0≤ Q2≤ 2[38]. In fact, this is the case, since the values of
the ratio mg/ obtained phenomenologically typically lies
in the interval mg/ ∈ [1.1, 2] [35–38,69–78].
3 QCD-based model 3.1 Eikonal representation
The correct calculation of high-energy hadronic interactions must be compatible with analyticity and unitarity constraints, where the latter is satisfied simply by means of eikonalized amplitudes. We adopt the following normalization for the elastic scattering amplitude:
A(s, t) = i ∞ 0 db b J0(qtb) 1− e−χ(s,b) , (6)
where s is the square of the total center-of-mass energy, b
is the impact parameter, qt2 = −t is the usual Mandelstam
invariant, with the complex eikonal function denoted by χ(s, b) = Re χ(s, b) + i Im χ(s, b)
≡ χR(s, b) + i χI(s, b). (7)
In this picture (s, b) = 1 − e−χ(s,b)is the profile
func-tion, which, by the shadowing property, describes the absorp-tion effects resulting from the opening of inelastic channels. In addition, in the impact parameter space and according to the unitarity condition of the scattering S-matrix it may be also written as 2Re (s, b) = | (s, b)|2+ 1− e−2χR(s,b) . (8)
Therefore, the scattering process cannot be uniquely inelas-tic since the elasinelas-tic amplitude receives contributions from both elastic and inelastic channels. In this representation
P(s, b) = e−2χR(s,b)can be defined as the probability that
neither hadron is broken up in a collision at a given b and s. Such an absorption factor is crucial to determine rapidity
gap survival probabilities in pp and ¯p p scattering at
high-energies, which in turn are crucial to disentangle inelastic diffractive (single and double) and central exclusive pro-cesses from the dominant minimum-bias (non-diffractive) cross section [79,80].
Within the eikonal representation, Eq. (6), the total cross
σt ot(s) = 4π ∞ 0 db b 1− e−χR(s,b)cosχ I(s, b) ; (9) ρ(s) = − ∞ 0 db b e−χR(s,b)sinχI(s, b) ∞ 0 db b 1− e−χR(s,b)cosχ I(s, b) . (10)
The eikonals for elastic pp and ¯p p scattering are
con-nected with crossing even (+) and odd (−) eikonals by
χpp¯p p(s, b) = χ+(s, b) ± χ−(s, b). (11)
Real and imaginary parts of the eikonals can be con-nected either by derivative dispersion relations (DDR) [81– 86] or Asymptotic Uniqueness (AU), which is based on the Phragmén–Lindelöff theorems [87,88] (see [89], appendixes B, C, D for a recent short review on these subjects). We have tested both methods and in what follows we present the results with the AU approach, also referred to as asymptotic prescriptions or real analytic amplitudes [88].
3.2 Semihard and soft contributions
The eikonal function is assumed to be the sum of the soft and the semihard (SH) parton interactions in the hadronic collision [90,91],
χ(s, b) = χso f t(s, b) + χS H(s, b), (12)
with each one related, in the general case, to the correspond-ing crosscorrespond-ing even and odd contributions:
χ±(s, b) = χso f t± (s, b) + χS H±(s, b). (13)
In what follows we specify the inputs for each one of the four aforementioned contributions to the eikonal.
3.2.1 Semihard contributions and the dynamical gluon mass
The fundamental basis of models inspired upon QCD, or also known as minijet models, is that the semihard scatterings of partons in hadrons are responsible for the observed increase of the total cross section. Here we assume a Pomeron dom-inance, represented by a crossing even contribution, namely we consider that the semihard odd component does not con-tribute with the scattering process,
χ−
S H = 0.
In respect to the even contribution, it follows from the QCD improved parton model. At leading order, this semihard eikonal can be factorized as
χS H+ (s, b) = 1
2WS H(s, b) σQC D(s), (14)
where WS H(s, b) is the overlap density distribution of
semi-hard parton scattering,σQC D denotes the cross section of
hard parton scattering in the region where pQCD can be
safely applied, namely above the cutoff Q2mi n.
We assume (as in previous studies [38]) that hard parton scattering configuration in the transverse plane of the colli-sion (in b-space) to be given by the Fourier–Bessel transform:
WS H(s, b; νS H) = 1 2π ∞ 0 dk⊥k⊥J0(k⊥b) [GS H(s, k⊥; νS H)] 2 = νS H2 96π(νS Hb) 3K 3(νS Hb), (15)
where GS H(s, k⊥) is the well-known dipole parametrization
GS H(s, k⊥; νS H) = ν2 S H k2⊥+ ν2 S H 2 , (16)
withνS H = νS H(s) taken as an energy dependent scale of the
dipole. Specifically, we assume a logarithmic dependence for νS H, namely:
νS H = ν1− ν2ln(s/s0), (17)
where ν1 andν2 are two free fit parameters and the scale
√
s0 = 5 GeV is fixed. Regarding this dependence of the form factor on the energy, though not being formally estab-lished in the context of QCD, it is truly supported by the wealth of accelerator data available (as we shall see in Sect. 4) and seems to us more realistic than taking a static par-tonic configuration in b-space. In addition, many other phe-nomenological models have been proposed in literature (see e.g. [92–99]), in which the energy dependence in form factors
play a crucial role in pp and ¯p p elastic scattering dynamics
and, therefore, in accurate descriptions of the data beyond
√
s∼10 GeV.
The dynamical contribution,σQC D(s), is calculated using
perturbative QCD as follows: σQC D(s) = i j 1 1+ δi j 1 0 d x1 1 0 d x2 ∞ Q2mi n d|ˆt|dˆσi j d|ˆt|(ˆs, ˆt) × fi/A(x1, |ˆt|) fj/B(x2, |ˆt|) ˆs 2− |ˆt| , (18)
where x1and x2are momentum fraction carried by partons
in the hadrons A and B, respectively, ˆs = x1x2s,|ˆt| ≡ Q2
stands for Mandelstam invariants of parton-parton scatterings
such as e.g. gg→ gg, qg → qg and gg → ¯qq (whose
par-tonic cross sections are given afterwards) and fi/A(x1, |ˆt|),
fj/B(x2, |ˆt|) are the parton distribution functions (PDFs) for
partons i and j. The indexes i, j = q, ¯q, g identify quark
(anti-quark) and gluon degrees of freedom and Q2mi n
rep-resent the minimum momentum transfer scale allowing for pQCD calculations of partonic hard scattering, obeying the constraint 2Q2mi n< 2|ˆt| < ˆs.
Concerning the differential cross section at elementary level, the major contribution at high energies are the ones initiated by gluons1
i. gluon–gluon elastic scattering, dˆσ dˆt(gg → gg) = 9π ¯α2s 2ˆs2 3− ˆtˆu ˆs2 − ˆs ˆu ˆt2 − ˆtˆs ˆu2 , (19) ii. quark-gluon elastic scattering,
dˆσ dˆt(qg → qg) = π ¯α2 s ˆs2 (ˆs 2+ ˆu2) 1 ˆt2 − 4 9ˆs ˆu , (20) iii. gluon fusion into a quark pair,
dˆσ dˆt(gg → ¯qq) = 3π ¯αs2 8ˆs2 (ˆt 2+ ˆu2) 4 9ˆtˆu − 1 ˆs2 , (21) with kinematical constraints imposed and connected with the
dynamical mass, namely: (i) ˆs + ˆt + ˆu = 4Mg2(Q2), for
gluon elastic scattering (gg → gg) and (ii) ˆs + ˆt + ˆu =
2Mg2(Q2) + 2Mq2(Q2) for gluon fusion (gg → ¯qq) and
quark-gluon scattering qg→ qg. Importantly, in what
fol-lows we assume the Cornwall’s dynamical gluon mass (in Euclidean space) [34], Eq. (3), with the infrared frozen effec-tive QCD charge, Eq. (4), to interpolate two QCD domains:
(i) Q2 ≈ 0, i.e. at infrared, where M2g freezes and the
gluons carries an effective bare mass, Mg2(0) = m2g; (ii)
Q2 m2g, 2, dynamical mass generation from
nontriv-ial vacuum structure becomes unimportant and perturbative QCD limit is achieved.
As discussed in Sect.2, recent phenomenology and lattice
studies support bare gluon masses in the range, mg: 300 −
700 MeV. Here we fix mg= 400 MeV
while also accounting, for completeness, the subdominant role of dynamical quark generation at high energies. We assume Mq(Q2) = m3q Q2+ m2 q , (22)
which also recovers the bare mass mq (with mq < mg) at
infrared and reaches the massless quark limit for Q2 m2q.
In all calculations we take mq = 250 MeV
1Despite the potential influence of soft gluon radiation at the initial state, such as discussed in [99] and references therein, we only consider the effects of gluon radiation in the Parton Distribution Functions, as following from DGLAP evolution.
as fixed scale. At last, as commented before, the complex
eikonalχS H+ (s, b) is determined through the asymptotic even
prescription s→ −is. The parametrization for σQC D(s), Eq.
(18), and the real and imaginary parts provided by the above prescription are presented and discussed in Appendix A. We notice that in case of WS H(s, b; νS H), Eq. (15), the prescription
results in a complex Bessel function of complex argument. 3.2.2 Soft contributions
The full even and odd soft contributions are based on the Regge–Gribov formalism and are constructed in accordance with Asymptotic Uniqueness (Phragmén–Lindelöff theo-rems). Assuming also leading even component, they are parametrized by χso f t+ (s, b) = 1 2W + so f t(b; μ + so f t)σ+(s), (23) χso f t− (s, b) = 1 2W − so f t(b; μ − so f t)σ −(s), (24) where σ+(s) = A +√B s/s0 eiπ/4+ C (s/s0) e−iπ/2 λ , (25) σ−(s) = De√−iπ/4 s/s0, (26)
denote analytical even and odd cross sections. Here λ =
0.12 and A, B, C and D are free fit parameters. Moreover,
the impact parameter structure derives from bidimensional Fourier transform of dipole form factors, namely:
Wso f t+ (b; μ+so f t) = 1 2π ∞ 0 dk⊥k⊥J0(k⊥b) G2di p(k⊥; μ+so f t) = (μ + so f t) 2 96π (μ + so f tb) 3 K3(μ+so f tb), (27) Wso f t− (b; μ−so f t) = (μ − so f t) 2 96π (μ − so f tb) 3 K3(μ−so f tb), (28) whereμ−so f t ≡ 0.5 GeV is a fixed parameter and μ+so f t a free fit parameter. As in the case of the SH form factor, the energy
scale is fixed at√s0= 5 GeV.
We notice that in the Regge-Gribov context, the soft even contribution consists of a Regge pole with intercept
α+R(0) = 1/2, a critical Pomeron and a single-pole Pomeron,
with interceptαP(0) = 1 + λ. The odd contribution is
asso-ciated with only a Regge pole, with interceptαR−(0) = 1/2.
Summarizing the model has 7 free fit parameters, 5 asso-ciated with the soft contribution, A, B, C, D, μ+so f t and only
2 with the semihard contribution,ν1andν2(fromνS H(s) in
WS H+(s, b)). In addition, 5 parameters are fixed: mg = 400
GeV, mq= 250 GeV, s0= 25 GeV2,μ−so f t = 0.5 GeV and
Table 1 Total cross section,σtot, andρ-parameter data recently mea-sured by TOTEM and ATLAS Collaborations at the LHC, but not com-piled in the PDG2018 review [100]
√
s (TeV) σtot[mb] ρ Collaboration Ref.
13 110.6 ± 3.4 – TOTEM [1] 110.3 ± 3.5 0.10 ± 0.01 TOTEM [2] 0.09 ± 0.01 8.0 − 0.12 ± 0.03 TOTEM [101] 102.9 ± 2.3 – 103.0 ± 2.3 – 96.07 ± 0.92 – ATLAS [103] 101.5 ± 2.1 – TOTEM [105] 101.9 ± 2.1 – 7.0 99.1 ± 4.3 − TOTEM [106] 95.35 ± 1.36 – ATLAS [104] 2.76 84.7 ± 3.3 – TOTEM [107]
4 Dataset and fit procedures
In the absence of ab initio theoretical QCD arguments to determine the parameters A, B, C, D, μ+so f t,ν1andν2, we resort to a fine-tuning fit procedure described in what follows.
As we are interested in the very high-energy behavior ofσt ot
andρ, we shall use only pp and ¯p p elastic scattering data.
Moreover, in order to test our QCD-based model in the t= 0
limit, we perform global fits that include exclusively forward data, given by Eqs. (9) and (10).
4.1 Dataset
Our dataset is compiled from a wealth of collider data on pp and¯p p elastic scattering, available in the Particle Data Group (PDG) database [100] as well as in the very recent papers of LHC Collaborations such as TOTEM [1,2,101,102] and ATLAS [103,104], which span a large c.m. energy range,
namely 10 GeV √s 13 TeV. For the sake of clarity
and completeness we furnish in Table1all the recent LHC
data onσt otandρ, still absent in the PDG2018 review. This
dataset totalizes 13 new data points on pp forward elastic scattering at high energies.
We call attention to the fact that we do not apply to this dataset, composed of 174 data points onσt ot¯p p,ppandρ¯p p,pp, any sort of selection or sieving procedure, which might intro-duce bias in the analysis.
4.2 Fit procedures
To provide statistical information on fit quality, we perform a best-fit analysis, furnishing as goodness of fit parameters
the chi-squared per degrees of freedom (χ2/ζ ) and the
cor-−6 10 10−5 10−4 10−3 10−2 10−1 1 x 0 0.5 1 1.5 2 2.5 ] -3 10× ) [ 2 xg(x,Q CT14 - Q=10 GeV CTEQ6L - Q=10 GeV MMHT - Q=10 GeV CT14 - Q=100 GeV CTEQ6L - Q=100 GeV MMHT - Q=100 GeV
Fig. 1 Gluon distribution function, xg(x, Q2), following from DGLAP evolution for PDFs, CT14, CTEQ6L and MMHT at Q= 10 GeV and Q= 100 GeV
responding integrated probability, P(χ2, ζ ) [108]. Since our model is highly nonlinear, numerical data reduction is called for. Despite the limitation of treating statistical and
system-atical uncertainties at the same foot, we apply theχ2/ζ tests
to our dataset with uncertainties summed in quadrature.2Our
fits are done using the TMINUIT class of the ROOT frame-work [111], through the MIGRAD algorithm. While the num-ber of calls of the MIGRAD routine may vary in the fits with PDFs CETQ6L, CT14 and MMHT, full convergence of the algorithm was always achieved. Moreover, all data
reduc-tions were performed with the intervalχ2− χmi n2 = 8.18,
which corresponds to 68.3 % of Confidence Level (1σ) [112]
in our case (7 free parameters).
Furthermore, in all fits performed we set the low energy
cutoff,√smi n = 10 GeV.
In the following we present our results, according to the choice of three distinct PDFs: CTEQ6L [42] (pre-LHC), CT14 [41] and MMHT [43] (fine-tuned with LHC data). In testing different PDFs we look for a better understanding of the impact of low-x parton dynamics in defining the very high-energy behavior ofσt ot¯p p,ppandρ¯p p,pp. For comparison and further discussion, the behavior of the gluon distribution
function in each PDF set is given in Figs. 1and2. In this
analysis all the TOTEM data onσt ot have been considered
as independent points in the data reductions. See Appendix B for a discussion on this respect.
2 For very recent applications of the frequentist and Bayesian approaches to high-energy elastic scattering data analysis see Refs. [109,110].
−8 10 10−7 10−6 10−5 10−4 10−3 10−2 10−1 1 x 0 2 4 6 8 10 ] -1 10× ) [ 2 xg(x,Q Q= 1.3 GeV CT14 CTEQ6L MMHT
Fig. 2 The same as Fig.1, but for the scale Q= 1.3 GeV
Table 2 Best fit parameters of the QCD-based model with PDFs CTEQ6L [42], CT14 [41] and MMHT [43]. Quality fit estimators, chi-squared per degree of freedom,χ2/ζ, and integrated probability, P(χ2; ζ ), are also furnished (where ζ = 167 specifies the number of degrees of freedom (dof) in each fit)
PDF CTEQ6L CT14 MMHT μ+so f t[GeV] 0.90±0.18 0.90±0.18 0.90±0.20 A [GeV−2] 101±11 88.1±9.7 93±11 B [GeV−2] 48±11 51.7±9.9 54±10 C [GeV−2] 16.0±6.8 27.3±5.8 19.6±6.7 μ−
so f t[GeV] 0.5 (fixed) 0.5 (fixed) 0.5 (fixed) D [GeV−2] 24.2±1.4 24.2±1.4 24.2±1.4 ν1[GeV] 1.63±0.20 1.70±0.22 1.46±0.21
ν2[GeV] 0.009±0.013 0.015±0.014 −0.007±0.013
χ2/ζ 1.285 1.304 1.259
P(χ2; ζ ) 7.6× 10−3 5.0× 10−3 1.3× 10−2
5 Results and discussion
The results for the free fit parameters, using each one of the three PDFs (CTEQ6L, CT14, MMHT) are displayed in
Table2, together with the statistical information on the data
reductions (reduced chi square and corresponding integrated
probability). The curves ofσt ot(s) and ρ(s) for the three
PDFs, compared with the experimental data, are shown in Fig.3. Predictions ofσt ot andρ for pp scattering at some
energies of interest are displayed in Table3.
From Fig.3 we see that, although the model provides a
quite good description of the forward data in the interval 10 GeV–8 TeV, the results at 13 TeV do not reach the error bars
of the TOTEM data onσt ot andρ.
Table 3 Predictions of our model using CT14 as a representative case √ s [TeV] σtot[mb] ρ 0.9 68.75 0.1338 2.76 82.91 0.1279 13 105.8 0.1194 14 107.0 0.1190 2 10 103 104 [GeV] s 40 60 80 100 120 140 [mb] tot σ pp accelerator pp accelerator pp cosmic rays CT14 MMHT CTEQ6L 4 10 90 95 100 105 110 115 120 10 102 103 104 [GeV] s −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 ρ pp accelerator pp accelerator CT14 MMHT CTEQ6L 4 10 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Fig. 3 Global 1σ-fit of total cross section, σtotpp/ ¯p pandρpp/ ¯p p param-eter. Best fit parameters and quality estimators are given in Table2
Before proceeding with further tests, let us discuss some physical aspects related to our results.
By showing the values in the Table2, we can see that the
parameterμ+so f t has, in general, the value 0.90 GeV. This
restriction is due to the fact that the inverse of both μ+so f t andμ−so f t parameters characterizes the range of these soft
interactions. Since the odd soft eikonalχso f t− (s, b) is more
sensitive to the longer-range trajectories,ρ and ω exchanges,
it is expected the inverse of the odd exchanges,(μ−so f t)−1, to
be larger than the inverse of the even (a2and f2) exchanges,
(μ+so f t)−1. Thus in our analysis we impose the reasonable condition 1 < μ+so f t/μ−so f t ≤ 1.8. Indeed, in all cases the parameterμ+so f t fall within the expected range.
In what concerns high-energy QCD dynamics, in QCD-based (s-channel) models like ours, the driving mechanism behind the rapid rise of the total cross section is linked to the growth with energy of low- ptjets (called minijets). This idea, while proposed many years ago, remains a powerful one in the scope of models of strong interactions at high-energies, as it provides a clear connection between perturbative QCD
and hadronic elastic observables, such asσt ot and ρ, in a
unitarized framework.
Those minijets arise from partonic interactions (mainly gluons) carring very small momentum fraction of their parent hadrons. On the one hand, from Eq. (18), we see that the
smallest product x1x2probed by the parton pair is
(x1x2)mi n= 2Q2mi n
s , (29)
which, taking Q2mi n 1 GeV2, yields(x1x2)mi n∼ 10−8at LHC13. In this case, if one of the partons has an especially
large fractional longitudinal momentum x∼ 1, the other one
has x ∼ 10−8. On the other hand, it is well-known that at
very low-x the PDF’s diverge, as gluon emissions – which naturally occur in any partonic process at high energies – are not suppressed by DGLAP evolution at higher momentum transferred. This behavior can be readily seen from Figs.
1and2where the gluon distribution function from parton
distributions CT14, CTEQ6L and MMHT are displayed at
the minimum scale Qmi n= 1.3 GeV and two higher scales,
Q= 10 GeV and 100 GeV. From these plots one may notice that MMHT grows faster than CT14 and CTEQ6L, specially
at low momentum scales, such as Qmi n= 1.3 GeV.
As matter of fact, very low-x gluons are the key ingredient to understand our results for various PDF’s, as shown in Fig. 3. Once the QCD cross section (18) is dominated by low-x partons, and gluon iniciated processes are the leading com-ponent of this cross section, one expects the magnitude of
σS H(s) calculated with MMHT to be larger than the
corre-sponding curves for CT14 and CTEQ6L at high energies. As
we show in Fig.9in Appendix A, that turns out to be exactly
the case.
In addition, looking for some insights into the formalism, it may be important to notice the effects of two phenomeno-logical inputs, one related to the soft even eikonal and the other to the semihard form factor. In the first case,χso f t+ (s, b) as given by Eq. (23), has a component which increases with the energy, namely the term with coefficient C. In the second
Table 4 Best-fit parameters of our model, obtained by fixing C= 0, withζ = 168 degrees of freedom
PDF CTEQ6L CT14 MMHT
μ+
so f t[GeV] 0.70±0.16 0.700±0.015 0.700±0.034 A [GeV−2] 111.31±0.70 117.86±0.62 109.9±0.77 B [GeV−2] 28.0±3.0 12.1±2.8 28.0±3.1 C [GeV−2] 0 (fixed) 0 (fixed) 0 (fixed) μ−so f t[GeV] 0.5 (fixed) 0.5 (fixed) 0.5 (fixed) D [GeV−2] 23.4±1.3 23.6±1.3 23.5±1.3 ν1[GeV] 1.68±0.12 1.54±0.13 1.41±0.12
ν2[GeV] 0.012±0.0084 0.0072±0.0091 −0.010±0.0087
χ2/ζ 1.338 1.836 1.385
P(χ2; ζ ) 2.3× 10−3 2.4× 10−10 7.1× 10−4
case, the dipole form factor GS H(s, k⊥; νS H), Eqs. (16) and (17), also depends on the energy through the logarithmic. The effect of these terms can be investigated by assuming either C = 0 or ν2 = 0 and re-fitting the dataset. Moreover, the efficience of the model for different choices of the datasets is also important to be checked. All these three variants are presented and discussed in the following three subsections. 5.1 Effect of the leading contribution inχso f t+ (s, b) The soft-even component of the eikonal, Eqs. (23) and (25), comprise a leading Pomeron contribution given by the power term in Eq. (25), with coefficient C. In order to investigate the relevance of this leading soft contribution at high energies in our global results, we present here a test in which this term
is excluded. Specifically, we fix C = 0 in Eq. (25) and refit
the dataset. The results of these fits are presented in Table4
and Fig.4.
In respect the statistical quality of the fits, comparison of
Tables2(C free parameter) and4(C = 0 fixed) shows that
the exclusion of this contribution results in a rather
unac-ceptable goodness of fit, sinceχ2/ζ increase to 1.3–1.8 and
P(χ2) decrease at least one order of magnitude. For
exam-ple, in case of CT14, from Table2(C free parameter),χ2/ζ =
1.304, P(χ2) = 5.0 × 10−3and from Table4(C= 0 fixed),
χ2/ζ = 1.836, P(χ2) = 2.4 × 10−10.
We conclude that, although not being the leading contri-bution at the highest energies, the single pole Pomeron in the soft component is important for an adequate fit result in
sta-tistical grounds. Moreover, the faster decrease ofρ observed
at LHC energies can related to the correlations among low-energy parameters such as A and B and high-low-energy ones, asν1andν2. In addition to the lower statistical significance of this fits, in comparison with the previous ones, the reduc-tion of B central values by a factor one-half and the large
2 10 103 104 [GeV] s 40 60 80 100 120 140 [mb] tot σ pp accelerator pp accelerator pp cosmic rays CT14 MMHT CTEQ6L 4 10 90 95 100 105 110 115 120 10 102 103 104 [GeV] s −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 ρ pp accelerator pp accelerator CT14 MMHT CTEQ6L 4 10 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Fig. 4 Global 1σ-fit to σpp/ ¯p p
tot andρpp/ ¯p pdata without rising soft terms inχso f t+ (s, b). Statistical information is provided in Table4
5.2 Effect of the energy dependence in the semihard form factor
Although not so usual in the present phenomenological con-text, one of the ingredients of the QCD-based model is the energy dependence embodied in the semihard form factor, Eqs. (15) and (16). As commented in our introduction, this assumption is associated with the possibility of a broadening of the spacial gluon distribution as the energy increases. In order to investigate the relevance of this assumption in our global results, we present here a test in which this energy
dependence is excluded. Specifically, we fixν2 = 0 in Eq.
(17), so thatνS H = ν1and refit the data set. As before, we
consider the three PDFs employed in this work. The results
of these fits are presented in Table5and Fig.5.
Table 5 Best-fit parameters of our model, obtained by fixingν2= 0, withζ = 168 degrees of freedom
PDF CTEQ6L CT14 MMHT μ+ so f t[GeV] 0.90±0.15 0.90±0.13 0.84±0.13 A [GeV−2] 106.0±8.1 93.7±7.8 92±14 B [GeV−2] 44.6±8.7 47.6±8.5 51±30 C [GeV−2] 12.8±4.7 23.7±4.5 17±24 μ−so f t[GeV] 0.5 (fixed) 0.5 (fixed) 0.5 (fixed) D [GeV−2] 24.2±1.3 24.2±1.3 24.0±1.8 ν1[GeV] 1.486±0.031 1.469±0.034 1.588±0.078
ν2[GeV] 0 (fixed) 0 (fixed) 0 (fixed)
χ2/ζ 1.304 1.362 1.263 P(χ2; ζ ) 4.9× 10−3 1.3× 10−3 1.2× 10−2 2 10 103 104 [GeV] s 40 60 80 100 120 140 [mb] tot σ pp accelerator pp accelerator pp cosmic rays CT14 MMHT CTEQ6L 4 10 90 95 100 105 110 115 120 10 102 103 104 [GeV] s −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 ρ pp accelerator pp accelerator CT14 MMHT CTEQ6L 4 10 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Fig. 5 Global 1σ-fit to σtotpp/ ¯p pandρpp/ ¯p pdata takingν2=0. Best fit parameters and quality estimators are given in Table5
Table 6 Best-fit parameters of our model with low-energy cut-off
√s
mi n= 5.0 GeV. The number of degrees of freedom isζ = 251
PDF CTEQ6L CT14 MMHT μ+ so f t[GeV] 0.90±0.16 0.90±0.16 0.90±0.17 A [GeV−2] 92.5±2.9 82.1±5.9 88.1±6.9 B [GeV−2] 58.3±4.5 60.0±4.1 62.1±4.4 C [GeV−2] 21.1±4.6 30.7±3.7 22.8±4.7 μ−so f t[GeV] 0.5 (fixed) 0.5 (fixed) 0.5 (fixed) D [GeV−2] 26.03±0.74 26.02±0.74 26.02±0.74 ν1[GeV] 1.70±0.20 1.76±0.21 1.49±0.21 ν2[GeV] 0.012±0.013 0.018±0.014 −0.006±0.014 χ2/ζ 1.451 1.464 1.430 P(χ2; ζ ) 3.8× 10−6 2.2× 10−6 8.8× 10−6 10 102 103 104 [GeV] s 40 60 80 100 120 140 pp accelerator pp accelerator pp cosmic rays CT14 MMHT CTEQ6L 4 10 90 95 100 105 110 115 120 10 102 3 10 104 [GeV] s −0.3 −0.2 −0.1 0 0.1 0.2 0.3 ρ pp accelerator pp accelerator CT14 MMHT CTEQ6L 4 10 0.08 0.1 0.12 0.14 0.16 0.18 0.2 [mb] tot σ
Fig. 6 Global 1σ-fit to σtotpp/ ¯p pandρpp/ ¯p pdata for low-energy cutoff
√s
mi n= 5 GeV
Table 7 Best-fit parameters of our model without ATLAS data and low-energy cutoff√smi n= 10 GeV. The number of degrees of freedom isζ = 165 PDF CTEQ6L CT14 MMHT μ+so f t[GeV] 0.90±0.18 0.90±0.17 0.90±0.20 A [GeV−2] 111±11 88.6±9.4 94±11 B [GeV−2] 48±10 51.3±9.6 54±10 C [GeV−2] 15.8±6.8 27.0±5.6 19.4±6.6 μ−
so f t[GeV] 0.5 (fixed) 0.5 (fixed) 0.5 (fixed) D [GeV−2] 24.2±1.4 24.2±1.4 24.2±1.4 ν1[GeV] 1.61±0.20 1.69±0.20 1.44±0.20
ν2[GeV] 0.0076±0.013 0.017±0.013 −0.009±0.012
χ2/ζ 1.278 1.261 1.251
P(χ2; ζ ) 9.3× 10−3 1.3× 10−2 1.6× 10−2
Comparison of Tables2(ν2free) and5(ν2= 0), shows that the statistical quality of the fits (χ2/ζ , P(χ2; ζ )) are similar,
but with a slight increase (decrease) inχ2/ζ (P(χ2; ζ )) in
case ofν2 = 0. From Figs.3 (ν2 free) and5 (ν2 = 0), we
notice distinct behaviors related to the use of the MMHT on one side and CT14/CTEQ6 on the other. This may be
related with the faster rise of σQC D(s) in case of MMHT,
as compared with the slower rise within CT14 and CTEQ6L
(see Fig.9in Appendix A).
We also notice that at 13 TeV, MMHT leads to the highest
values of bothσt otandρ. For CT14 and CTEQ6L, although
theρ results reach the upper error bar, those for σt ot lie far
below the lower error bar. 5.3 Changing the data-set
Here we develop two tests on the efficiency of the QCD-based model related to two different choices of the dataset. In the first test the low-energy cutoff is lowered from 10 GeV down to 5 GeV and in the second test the ATLAS data at 7 and 8 TeV are not included in the dataset. We present the results obtained with the three PDFs. Since the results are similar to those presented in the main text with our standard dataset, we focus the discussion on those obtained with the PDF CT14.
5.3.1 Low-energy cutoff down to 5 GeV
By lowering the energy cutoff to 5 GeV, we add 85 points for
σt ot andρ in the dataset. The result of the fit is displayed in
Table6and Fig.6, indicating, within CT14,χ2/ζ = 1.464,
forζ = 251 and P(χ2; ζ ) = 2.2 × 10−6. Our results with
cutoff at 10 GeV are shown in Fig.3 and Table2(CT14)
and in this case,χ2/ζ = 1.304, for ζ = 167 and P(χ2; ζ ) =
2 10 103 104 [GeV] s 40 60 80 100 120 140 [mb] tot σ pp accelerator pp accelerator pp cosmic rays CT14 MMHT CTEQ6L 4 10 90 95 100 105 110 115 120 2 10 103 104 [GeV] s −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 ρ pp accelerator pp accelerator CT14 MMHT CTEQ6L 4 10 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Fig. 7 Global 1σ-fit to σtotpp/ ¯p pandρpp/ ¯p pdata without the ATLAS measurements, for low-energy cutoff√smi n= 10 GeV
Table 8 Best-fit parameters of our model without ATLAS data and low-energy cutoff√smi n= 5.0 GeV. The number of degrees of freedom isζ = 249 PDF CTEQ6L CT14 MMHT μ+ so f t[GeV] 0.90±0.15 0.90±0.14 0.90±0.16 A [GeV−2] 92.9±6.8 82.3±6.0 88.4±6.8 B [GeV−2] 58.1±4.5 59.8±4.2 61.9±4.4 C [GeV−2] 20.9±4.5 30.5±3.8 22.6±4.6 μ−so f t[GeV] 0.5 (fixed) 0.5 (fixed) 0.5 (fixed) D [GeV−2] 26.03±0.74 26.02±0.73 26.02±0.74 ν1[GeV] 1.70±0.19 1.75±0.20 1.50±0.20 ν2[GeV] 0.014±0.012 0.019±0.013 −0.004±0.013 χ2/ζ 1.422 1.438 1.395 P(χ2; ζ ) 1.3× 10−5 7.0× 10−6 3.7× 10−5 10 102 3 10 104 [GeV] s 40 60 80 100 120 140 [mb] tot σ pp accelerator pp accelerator pp cosmic rays CT14 MMHT CTEQ6L 4 10 90 95 100 105 110 115 120 10 102 103 104 [GeV] s −0.3 −0.2 −0.1 0 0.1 0.2 0.3 ρ pp accelerator pp accelerator CT14 MMHT CTEQ6L 4 10 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Fig. 8 Global 1σ-fit to σtotpp/ ¯p pandρpp/ ¯p pdata without the ATLAS measurements, for maximum-energy cutoff√smax= 13 TeV and low-energy cutoff√smi n= 5 GeV
Although the integrated probability decreases three orders
of magnitude for√smi n= 5.0 GeV, from Figs.3and6, we
see that the visual description of the data is quite good and the quality of the fit is reasonable for this data set (without any sieve procedure), showing that the model can cover effi-ciently the whole region 5 GeV–8 TeV. Still, the problem of
simultaneously fittingσt ot andρ at 13 TeV (within
uncer-tainties) remains.
5.3.2 Fits without the ATLAS data
It is well known the discrepancies between the TOTEM and
ATLAS data onσt ot at 7 and 8 TeV [113]. Here we present
which the ATLAS data are not included in the data set. The results are presented in Table7, Fig.7(√smi n= 10 GeV) and Table8, Fig.8(√smi n= 5 GeV).
Comparing the results of Table7and Fig.7with the ones in
Table2and Fig.3(with complete dataset) we see that,
with-out the ATLAS data, the integrated probability increases as a consequence of the aforementioned discrepancies. On the one hand, it is interesting to note that the exclusion of the
ATLAS data leads to a better description ofσt ot at LHC13,
both in visual and statistical grounds, for PDFs CT14 and CTEQ6L. On the other, MMHT prediction gives a lower
cross section at 13 TeV, while providing a smallerρ. Such
behavior can be understood in the light of PDFs small-x extrapolation, as we discuss in the following.
A final interesting remark about these results lies in the
fact that, by lowering the energy cutoff√smi n to 5.0 GeV
improves the descriptionσt ot at LHC13 for all PDFs, as we
note that in such case, the uncertainties in MMHT low-energy parameters ( A and B) are reduced. Yet, incompatibility of the
model results with TOTEM’s measurements ofσt otandρ at
LHC13 are evident in these cases.
6 Conclusions
In this paper we have discussed recent studies on forward pp and ¯p p elastic scattering within an eikonal QCD-based for-malism. The model combines the perturbative parton-model approach (to model the semihard interactions among par-tons), with a Regge-inspired model (to describe the under-lying soft interactions) and brings up information about the infrared properties of QCD by considering the possibility that the nonperturbative dynamics of QCD generate an effective charge.
We have presented a phenomenological analysis under-taken to improve the understanding of elastic processes tak-ing place in the LHC. We address this issue by means of a model involving only even-under-crossing amplitudes at very high energies. As a result, we see that the QCD-based model allows us to describe the forward scattering quantities
σt ot andρ from √s = 10 GeV to 8 TeV in a quite
satis-factory way, but not the TOTEM measurements at 13 TeV simultaneously.
Our analysis, which follows a previous short letter [40], explores in detail the various effects that could be important in the global fits, in special the use of three different PDFs (CT14,CTEQ6L and MMHT), investigating not only the dif-ference and similarities among them, but also the effect of being pre or post LHC distributions.
On general grounds, from a statistical viewpoint, the present results demonstrate an overall satisfactory agreement
of all PDFs withσt ot andρ data over a wide range of
ener-gies. However, specifically at√s= 13 TeV, our results for ρ
(σt ot) are greater (lower) than the TOTEM measurements. We
understand that the inclusion of a crossing-odd elastic term in the scattering amplitude may improve the description of the forward data at high energies. Such a result might be an indi-cation that an Odderon does indeed have an important role in
the soft and/or semihard interactions at LHC energies.3We
are presently investigating the subject.
Acknowledgements We thank Victor Gonçalves for useful discus-sions. This research was partially supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under the Grants 141496/2015-0 and 155628/2018-6, by the project INCT-FNA Proc. No. 464898/2014-5, by the PEDECIBA program, and by the ANII-FCE-126412 project.
Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theo-retical paper based on public experimental data, therefore without any additional data associated.]
Open Access This article is licensed under a Creative Commons Attri-bution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indi-cated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permit-ted use, you will need to obtain permission directly from the copy-right holder. To view a copy of this licence, visithttp://creativecomm ons.org/licenses/by/4.0/.
Funded by SCOAP3.
Appendix A: Parametrization forσQC D(s)
One of the most important ingredient of the QCD-based model is the even-under-crossing partonic cross-section
σQC D(s), given by Eq. (18). Here we present the details of
the evaluation of this quantity, using PDFs: CTEQ6L , CT14 and MMHT. Some additional results are also presented and discussed. The evaluation is based on the steps that follow.
First we consider the complex analytic parametrization σQC D(s) = b1+ b2eb3[X(s)] 1.01 b4 + b5eb6[X(s)] 1.05 b7 + b8eb9[X(s)] 1.09 b10 , (A1)
3 It is worth noting that some recent well known phenomenological approaches, with even-under-crossing dominance and without a satis-factory simultaneous description of the TOTEM data at 13 TeV, indi-cate a different interpretation. For example, Donnachie and Landshoff, within the Regge approach, obtain for rho at 13 TeV the value 0.14 and conclude that “there is no strong case for the presence of an odderon contribution to forward scattering” [27]. Durand and Ha in the context of a QCD-based model, employ the sieve process proposed by Martin Block [114], which excludes the rho measurements at 13 TeV from the dataset as outliers [26].
Table 9 Fit results of the ReσQC Din Eqs. (1) and (2) to the actual data (see text) PDF CTEQ6L CT14 MMHT b1[GeV−2] 97.005 100.220 95.284 b2[GeV−2] 0.280× 10−1 0.434× 10−1 0.372 b3 1.699 1.274 0.600 b4 1.736 1.919 2.496 b5[GeV−2] −0.149 × 10−5 0.122× 10−7 −0.255 × 10−5 b6 14.140 14.050 14.281 b7 0.319 0.504 0.281 b8[GeV−2] 0.836× 10−1 3.699× 103 0.909 b9 3.813 −80.280 4.290 b10 0.810 −2.632 0.673 10 102 3 10 104 5 10 [GeV] s 0 2000 4000 6000 8000 10000 12000 14000 ] -2 (s) [GeV QCD σ Complex Parametrization = 1.3 GeV min Q = 400 MeV g m (s) QCD σ (CTEQ6L) Re (s) QCD σ (CTEQ6L) -Im (s) QCD σ (CT14) Re (s) QCD σ (CT14) -Im (s) QCD σ (MMHT) Re (s) QCD σ (MMHT) -Im
Fig. 9 Real and imaginary parts of the complexσQC Dfor each PDF. Small dots represent theoretical LO calculations from Eq. (18) – roughly 30 points for each PDF. Solid curves correspond to ReσQC D(s) fit, Eqs. (A1) and (A2), to the data, represent by the dots, with less than 5% per datum j= 1, . . . , 30. Dashed curves give Im σQC D(s), as calculated from Eqs. (A1) and (A2), using fit parameters furnished in Table9
where b1, . . . , b10are free fit parameters and
X(s) = ln ln(−is) (A2)
provides the adequate complex and even character of the
analytic function through the substitution s→ −is, leading
to ReσQC D(s) and Im σQC D(s).
Next, by means of Eq. (18) and using the three distinct PDFs, we generate around 30 points for each one of these parton distributions, which are then fitted by the ReσQC D(s), with less than 1% error. With the values of the free fit
parame-ters determined for each PDF, the corresponding ImσQC D(s)
are evaluated.
Table 10 Fit results by enlarging the uncertainties in the TOTEM data onσtot at 8 TeV and 13 TeV by factors
√
5 and√2, respectively. See text PDF CTEQ6L CT14 MMHT μ+ so f t[GeV] 0.90±0.19 0.90±0.19 0.90±0.20 A [GeV−2] 101±11 87.8±9.8 93±10 B [GeV−2] 48±10 51.9±9.9 55±10 C [GeV−2] 16.0±6.9 27.4±5.8 19.6±6.5 μ−so f t[GeV] 0.5 (fixed) 0.5 (fixed) 0.5 (fixed) D [GeV−2] 24.2±1.4 24.2±1.4 24.2±1.4 ν1[GeV] 1.61±0.21 1.70±0.23 1.44±0.20
ν2[GeV] 0.007±0.013 0.013±0.014 −0.010±0.013
χ2/ζ 1.178 1.193 1.156
P(χ2; ζ ) 5.8× 10−2 4.5× 10−2 8.1× 10−2
For CTEQ6L, CT14 and MMHT we display in Table9
the best-fit parameters bi, i = 1, · · · , 10 and in Fig.9the
dependencies of ReσQC D(s) and Im σQC D(s).
From the figure, we see in all cases the steep rise of the
partonic cross-sections with the energy. For example at√s=
10 TeV, most results lie around 580 mb. Notice, however, that this rise is tamed in the physical cross-sections, since we have an eikonalized model.
We note that among the PDFs post-LHC, MMHT led to the
fastest rise of both ReσQC D(s) and Im σQC D(s) and CT14
led to the slowest rise. The results with CTEC6L (pre-LHC) lie between these two cases.
The extreme fast rise ofσQC D(s) in case of MMHT, may
be the responsible for the decrease ofρ at the LHC region
(see Fig.3).
Appendix B: Comments on fit procedure and indepen-dent points
The TOTEM results forσt ot at 8 TeV (5 points) and 13 TeV
(2 points) are displayed in Table1. These measurements have
been obtained through different methods, different physical assumptions, different mathematical modeling and even dif-ferent intervals in momentum transfer, as described in the quoted references. However, it might be argued that, at each energy, these results correspond to different analyses based on the same differential cross section data and for that reason should not be considered independent points, as assumed in our analysis.
In order to test the influence of this assumption we have developed new data reductions by multiplying the uncertain-ties in each datum by the square root of the number of points
at each energy, namely√5 at 8 TeV and√2 at 13 TeV. The
2 10 103 104 [GeV] s [GeV] s 40 60 80 100 120 140 [mb] tot σ pp accelerator pp accelerator pp cosmic rays CT14 MMHT CTEQ6L 4 10 90 95 100 105 110 115 120 10 102 3 10 104 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 ρ pp accelerator pp accelerator CT14 MMHT CTEQ6L 4 10 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Fig. 10 Fits toσtotpp/barppandρpp/barppdata with statistical weighting factors√5 and√2 in the TOTEM data onσtotat 8 TeV and 13 TeV, respectively. See text
Let us discuss these results by comparing with those obtained under our assumption of independent points, namely
Table2and Fig.3. In what concerns the values of the free
parameters, we see that in all cases they are consistent within the uncertainties and the behaviors of the curves present only slight differences and do not change the scenario: the TOTEM
data at 13 TeV onσt otandρ are not simultaneously described.
In respect the statistical quality of the fits, in case of assuming
dependent points, theχ2/ζ decreases from ∼ 1.3 to ∼ 1.2
and the integrated probabilities increase around one order of magnitude. That, however, seems not to be an indication of best fit quality, but a consequence of the increments in the uncertainties.
We conclude that, in the present case, the two proce-dures led to numerical, statistical and physical results that are equivalent.
References
1. TOTEM, G. Antchev et al., Eur. Phys. J. C 79, 103 (2019) 2. TOTEM, G. Antchev et al., Eur. Phys. J. C 79, 785 (2019) 3. L. Lukaszuk, B. Nicolescu, Lett. Nuovo Cim. 8, 405 (1973) 4. C. Ewerz,arXiv:hep-ph/0511196(2005)
5. C. Ewerz,arXiv:hep-ph/0306137(2003)
6. J.R. Forshaw, D.A. Ross, Camb. Lect. Notes Phys. 9, 1 (1997) 7. S. Donnachie, H.G. Dosch, O. Nachtmann, P. Landshoff, Camb.
Monogr. Part. Phys. Nucl. Phys. Cosmol. 19, 1 (2002) 8. E. Martynov, B. Nicolescu, Phys. Lett. B 778, 414 (2018) 9. E. Martynov, B. Nicolescu, Phys. Lett. B 786, 207 (2018) 10. E. Martynov, B. Nicolescu, Eur. Phys. J. C 79, 461 (2019) 11. E. Martynov, B. Nicolescu,arXiv:1811.07635[hep-ph] (2018) 12. T. Csörg˝o, R. Pasechnik, A. Ster, Eur. Phys. J. C 79, 62 (2019) 13. T. Csörg˝o, R. Pasechnik, A. Ster, Lévy imaging of elastic
hadron-hadron scattering: Odderon and inner structure of the proton, in Diffraction and Low-x 2018 (Difflowx2018) Reggio Calabria, Italy, August 26–September 1, 2018, (2019), 1902.00109 14. V.P. Gonçalves, P.V.R.G. Silva, Eur. Phys. J. C 79, 237 (2019) 15. L.A. Harland-Lang, V.A. Khoze, A.D. Martin, M.G. Ryskin, Phys.
Rev. D 99, 034011 (2019).arXiv:1811.12705
16. V.P. Gonçalves, Eur. Phys. J. C 79, 408 (2019)
17. V.A. Khoze, A.D. Martin, M.G. Ryskin, Phys. Lett. B 784, 192 (2018)
18. E. Gotsman, E. Levin, I. Potashnikova, Phys. Lett. B 786, 472 (2018)
19. E. Gotsman, E. Levin, I. Potashnikova, Phys. Lett. B 781, 155 (2018)
20. P. Lebiedowicz, O. Nachtmann, A. Szczurek, EPJ Web Conf. 206, 06005 (2019).arXiv:1812.00138[hep-ph]
21. I. Szanyi, N. Bence, L. Jenkovszky, J. Phys. G 46, 055002 (2019) 22. Y.M. Shabelski, A.G. Shuvaev, Eur. Phys. J. C 78, 497 (2018) 23. M. Broilo, E.G.S. Luna, M.J. Menon,arXiv:1803.06560[hep-ph]
(2018)
24. M. Broilo, E.G.S. Luna, M.J. Menon, Phys. Lett. B 781, 616 (2018)
25. M. Broilo, E. Luna, M. Menon, Phys. Rev. D 98, 074006 (2018) 26. L. Durand, P. Ha, Phys. Rev. D 99, 014009 (2019).
arXiv:1810.11325
27. A. Donnachie, P.V. Landshoff, Phys. Lett. B 798, 135008 (2019).
arXiv:1904.11218
28. P. Lebiedowicz, O. Nachtmann, A. Szczurek, Phys. Rev. D 98, 014001 (2018)
29. G. Pancheri, S. Pacetti, Y. Srivastava, Phys. Rev. D 99, 034014 (2019)
30. A.K. Kohara, E. Ferreira, M. Rangel, Phys. Lett. B 789, 1 (2019) 31. J.R. Cudell, O.V. Selyugin (2019).arXiv:1901.05863[hep-ph] 32. D. Cline, F. Halzen, J. Luthe, Phys. Rev. Lett. 31, 491 (1973) 33. G. Pancheri, Y.N. Srivastava, Phys. Lett. B 182, 199 (1986) 34. J.M. Cornwall, Phys. Rev. D 26, 1453 (1982)
35. E.G.S. Luna, A.F. Martini, M.J. Menon, A. Mihara, A.A. Natale, Phys. Rev. D 72, 034019 (2005)
36. E.G.S. Luna, A.A. Natale, Phys. Rev. D 73, 074019 (2006) 37. D.A. Fagundes, E.G.S. Luna, M.J. Menon, A.A. Natale, Nucl.
Phys. A 886, 48 (2012)
38. C.A.S. Bahia, M. Broilo, E.G.S. Luna, Phys. Rev. D 92, 074039 (2015)
39. C.A.S. Bahia, M. Broilo, E.G.S. Luna, J. Phys: Conf. Ser. 706, 052006 (2016).arXiv:1508.07359[hep-ph]
40. M. Broilo, D.A. Fagundes, E.G.S. Luna, M.J. Menon, Phys. Lett. B 799, 135047 (2019).arXiv:1904.10061
41. S. Dulat et al., Phys. Rev. D 93, 033006 (2016) 42. J. Pumplin et al., JHEP 07, 012 (2002)
43. L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Eur. Phys. J. C 75, 204 (2015)
44. I.C. Cloet, C.D. Roberts, Prog. Part. Nucl. Phys. 77, 1 (2014) 45. A.C. Aguilar, D. Binosi, J. Papavassiliou, Front. Phys. (Beijing)
11, 111203 (2016)
46. C.D. Roberts, Few Body Syst. 58, 5 (2017)
47. A. Cucchieri, T. Mendes, PoS LATTICE2007, 297 (2007) 48. A. Cucchieri, T. Mendes, Phys. Rev. Lett. 100, 241601 (2008) 49. A. Cucchieri, T. Mendes, Phys. Rev. D 81, 016005 (2010) 50. P.O. Bowman et al., Phys. Rev. D 76, 094505 (2007)
51. I.L. Bogolubsky, E.M. Ilgenfritz, M. Muller-Preussker, A. Stern-beck, Phys. Lett. B 676, 69 (2009)
52. O. Oliveira, P.J. Silva, PoS LAT2009, 226 (2009)
53. A. Ayala, A. Bashir, D. Binosi, M. Cristoforetti, J. Rodriguez-Quintero, Phys. Rev. D 86, 074512 (2012)
54. P. Bicudo, D. Binosi, N. Cardoso, O. Oliveira, P.J. Silva, Phys. Rev. D 92, 114514 (2015)
55. L. von Smekal, R. Alkofer, A. Hauck, Phys. Rev. Lett. 79, 3591 (1997)
56. A.C. Aguilar, J. Papavassiliou, JHEP 12, 012 (2006)
57. C.S. Fischer, J.M. Pawlowski, Phys. Rev. D 75, 025012 (2007) 58. A.C. Aguilar, D. Binosi, C.T. Figueiredo, J. Papavassiliou, Eur.
Phys. J. C 78, 181 (2018)
59. C.S. Fischer, J.M. Pawlowski, A. Rothkopf, C.A. Welzbacher, Phys. Rev. D 98, 014009 (2018)
60. A. Deur, S.J. Brodsky, G.F. de Teramond, Prog. Part. Nucl. Phys. 90, 1 (2016)
61. D. Zwanziger, Phys. Rev. D 65, 094039 (2002) 62. H. Gies, Phys. Rev. D 66, 025006 (2002) 63. J. Braun, H. Gies, JHEP 06, 024 (2006) 64. D. Zwanziger, Phys. Rev. D 69, 016002 (2004)
65. D. Dudal, J.A. Gracey, S.P. Sorella, N. Vandersickel, H. Ver-schelde, Phys. Rev. D 78, 065047 (2008)
66. F. Gao, S.-X. Qin, C.D. Roberts, J. Rodriguez-Quintero, Phys. Rev. D 97, 034010 (2018)
67. C.D. Roberts, A.G. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994)
68. R. Alkofer, L. von Smekal, Phys. Rept. 353, 281 (2001) 69. E.G.S. Luna, Phys. Lett. B 641, 171 (2006)
70. E.G.S. Luna, A.L. dos Santos, A.A. Natale, Phys. Lett. B 698, 52 (2011)
71. V. Sauli, J. Phys. G 39, 035003 (2012)
72. S. Jia, F. Huang, Phys. Rev. D 86, 094035 (2012) 73. P. Lipari, M. Lusignoli, Eur. Phys. J. C 73, 2630 (2013) 74. A.V. Giannini, F.O. Duraes, Phys. Rev. D 88, 114004 (2013) 75. A.V. Sidorov, O.P. Solovtsova, Nonlinear Phenom. Complex Syst.
16, 397 (2013)
76. P. Allendes, C. Ayala, G. Cvetiˇc, Phys. Rev. D 89, 054016 (2014) 77. P.C. Beggio, E.G.S. Luna, Nucl. Phys. A 929, 230 (2014) 78. G. Cvetic, Phys. Rev. D 89, 036003 (2014)
79. D.A. Fagundes, A. Grau, G. Pancheri, O. Shekhovtsova, Y.N. Sri-vastava, Phys. Rev. D 96, 054010 (2017)
80. V.A. Khoze, A.D. Martin, M.G. Ryskin, J. Phys. G 45, 053002 (2018)
81. J.B. Bronzan, G.L. Kane, U.P. Sukhatme, Phys. Lett. B 49, 272 (1974)
82. M.M. Block, R.N. Cahn, Rev. Mod. Phys. 57, 563 (1985) 83. R.F. Avila, M.J. Menon, Nucl. Phys. A 744, 249 (2004) 84. R.F. Avila, M.J. Menon, Br. J. Phys. 37, 358 (2007)
85. R.F. Avila, E.G.S. Luna, M.J. Menon, Br. J. Phys. 31, 567 (2001) 86. R.F. Avila, E.G.S. Luna, M.J. Menon, Phys. Rev. D 67, 054020
(2003)
87. R. Eden, High Energy Collisions of Elementary Particles, 1 edn. (Cambridge University Press, Cambridge)
88. M.M. Block, Phys. Rept. 436, 71 (2006)
89. D.A. Fagundes, M.J. Menon, P.V.R.G. Silva, Int. J. Mod. Phys. A 32, 1750184 (2017)
90. L. Durand, P. Hong, Phys. Rev. Lett. 58, 303 (1987) 91. L. Durand, H. Pi, Phys. Rev. D 40, 1436 (1989) 92. B. Carreras, J.N.J. White, Nucl. Phys. B 42, 95 (1972) 93. J.N.J. White, Nucl. Phys. B 51, 23 (1973)
94. M.J. Menon, Nucl. Phys. Proc. Suppl. B 25, 94 (1992)
95. R.J.M. Covolan, P. Desgrolard, M. Giffon, L.L. Jenkovszky, E. Predazzi, Z. Phys. C 58, 109 (1993)
96. M.J. Menon, Can. J. Phys. 74, 594 (1996)
97. P. Lipari, M. Lusignoli, Phys. Rev. D 80, 074014 (2009) 98. D.A. Fagundes, A. Grau, S. Pacetti, G. Pancheri, Y.N. Srivastava,
Phys. Rev. D 88, 094019 (2013)
99. D.A. Fagundes, A. Grau, G. Pancheri, Y.N. Srivastava, O. Shekhovtsova, Phys. Rev. D 91, 114011 (2015)
100. Particle Data Group, M. Tanabashi et al., Phys. Rev. D 98, 030001 (2018)
101. TOTEM, G. Antchev et al., Eur. Phys. J. C 76, 661 (2016) 102. TOTEM, G. Antchev et al., CERN-EP-2018-341,
TOTEM-2018-002,arXiv:1812.08610[hep-ex] (2018)
103. ATLAS, M. Aaboud et al., Phys. Lett. B 761, 158 (2016) 104. ATLAS, G. Aad et al., Nucl. Phys. B 889, 486 (2014) 105. TOTEM, G. Antchev et al., Nucl. Phys. B 899, 527 (2015) 106. TOTEM, G. Antchev et al., EPL 101, 21004 (2013)
107. F. Nemes, Recent results from 2.76 and 13 TeV p+p collisions from the TOTEM experiment at CERN LHC, in talk in Zimányi School 2018, Budapest, Hungary, 03-07.12.2018, February 14th, (2019)
108. P. Bevington, D. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 3 ed. (McGraw-Hill Education) 109. R. Orava, O.V. Selyugin,arXiv:1804.05201[hep-ex] (2018) 110. O.V. Selyugin, J.R. Cudell, Odderon, HEGS model and LHC data,
in Diffraction and Low-x 2018 (Difflowx2018) Reggio Calabria, Italy, August 26-September 1, 2018, (2018).arXiv:1810.11538
111. R. Brun, F. Rademakers, Nucl. Instrum. Methods A 389, 81 (1997) 112. F. James, M. Roos, Comput. Phys. Commun. 10, 343 (1975) 113. D.A. Fagundes, M.J. Menon, P.V.R.G. Silva, Nucl. Phys. A 966,
185 (2017)
114. M.M. Block, Nucl. Instrum. Methods. A 556, 308 (2006).
